Here, \(f_{U_{t+1|1:t}}\)  is the density of \((U_{t+1}|U_1,\ldots ,U_t)\) , which is not uniform on \([0,1]\)  (whereas the marginal distribution of \(U_{t+1}\)  is uniform on \([0,1]\) ).
This conditional density can be used to form predictions from the copula model. It can also be used in likelihood-based estimation because
\(f_Y(\text{$y$})=\prod _{t=2}^T\left\lbrace  f_{U_{t|1:t-1}}(u_{t}|u_1,\ldots ,u_{t-1})f_{Y_{t}}(y_{t})\right\rbrace f_{Y_1}(y_1)\) , with
\(\text{$y$}=(y_1,\ldots ,y_T)^\top \) ,
which can be computed
efficiently for many choices of copula \(c_{1:T}\) .
In drawable vine
copulas (D-vines) \(f_{U_{t+1|1:t}}\)  is further decomposed into a product of bivariate
copulas called “pair-copulas” [1]}, allowing for a flexible
representation of the serial dependence structure, as discussed by [2]},
[3]}, [4]}, [5]}, [6]} and others.
